Natural units
See also: Dimensional analysis and Nondimensionalization Fundamental units: *Space *Time (In relativity time has units of imaginary distance) *Mass *Charge The field surrounding a charge holds energy. The total energy is: : E \propto \frac{q^2}{r} Therefore: : {\color{red}E*\frac{r}{q^2} } = constant Coulomb's law states that: : F={\color{red}k_e} \frac{q_1 q_2}{d^2} where: * k_e is the Coulomb constant which is the "stiffness" of space. (If space were stiffer then c would be faster.) The Coulomb constant has units of Energy*distance/charge2 which gives: : F = {\color{red}F * d \frac{d}{q^2}} \frac{q_1 q_2}{d^2} Newton's law of universal gravitation states that: : F = G \frac{m_1 m_2}{d^2} where: : F = \frac{F * d * d}{m^2} \frac{m_1 m_2}{d^2} But its probably better to say that: : a = \frac{d}{t^2} = G \frac{m}{d^2}\ Setting the force between two electrons equal to the force between two unknown masses we get: : G \frac{m_1 m_2}{d^2} = k_e\frac{q_1 q_2}{d^2} Solving we get m = 1.859 * 10-6 g = \sqrt{\alpha} planck masses = 1 Stoney mass The Schwarzschild radius of a Stoney mass is 2 Stoney lengths. Natural units From Wikipedia:natural units: In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. The surface area of a sphere 4 \pi r^2 In Lorentz–Heaviside units (rationalized units), Coulomb's law is: * F=\frac{q_1 q_2}{r^2} \frac{1}{4 \pi} In Gaussian units (non-rationalized units), Coulomb's law is: * F=\frac{q_1 q_2}{r^2} Planck units are defined by : = ''k = 1}}, Stoney units are defined by: : = e'' = ''k = 1}}, Hartree atomic units are defined by: : = ''ħ = k'' = ''k = 1}} : }} Rydberg atomic units are defined by: : }} = 2''m'' = ħ'' = ''k = k'' = 1}} : }} Quantum chromodynamics (QCD) units are defined by: : = ''ħ = k'' = 1}} 'Natural units' generally means: : = 1}}. where: * is the speed of light, * is the reduced Planck constant, * is the gravitational constant, * }} is the Coulomb constant, * is the Boltzmann constant * is the elementary charge, Base units Summary table From Wikipedia:natural units: where: * is the dimensionless fine-structure constant * }} is the dimensionless gravitational coupling constant * is dimensionless proton-to-electron mass ratio Fine-structure constant From Wikipedia:Fine-structure constant: The Fine-structure constant, , in terms of other fundamental physical constants: : \alpha = \frac{1}{4 \pi \varepsilon_0} \frac{e^2}{\hbar c} = \frac{\mu_0}{4 \pi} \frac{e^2 c}{\hbar} = \frac{k_\text{e} e^2}{\hbar c} = \frac{c \mu_0}{2 R_\text{K}} = \frac{e^2}{4 \pi}\frac{Z_0}{\hbar} where: * is the elementary charge * is the mathematical constant pi * is the reduced Planck constant * is the speed of light in vacuum * is the electric constant or permittivity of free space * is the magnetic constant or permeability of free space * is the Coulomb constant * is the von Klitzing constant * is the vacuum impedance or impedance of free space Gravitational coupling constant From Wikipedia:Gravitational coupling constant: The Gravitational coupling constant, , is typically defined in terms of the gravitational attraction between two electrons. More precisely, : \alpha_\mathrm{G} = \frac{G m_\mathrm{e}^2}{\hbar c} = \left( \frac{m_\mathrm{e}}{m_\mathrm{P}} \right)^2 \approx 1.751751596 \times 10^{-45} where: * is the gravitational constant * is the electron rest mass * is the speed of light in vacuum * is the reduced Planck constant * is the Planck mass Boltzmann constant From Wikipedia:Boltzmann constant: The Boltzmann constant, , is a scaling factor between macroscopic (thermodynamic temperature) and microscopic (thermal energy) physics. Macroscopically, the ideal gas law states: : k_B T = P \frac{V}{n} where: * is the Boltzmann constant * is the temperature * is the pressure * is the volume * is the number of molecules of gas. The pressure exerted on one face of a cube of length by a single particle of mass and velocity v = \sqrt{v_x + v_y + v_z} is: : pressure = \frac{force}{area} = \frac{\frac{momentum}{time}}{d^2} = \frac{\frac{2 m v_x}{2 d / v_x}}{d^2} = \frac{m v_x^2}{d^3} = \frac{2 E_x}{V_0} where: * is the volume occupied by a single particle. * = * is the velocity perpendicular to the face **Twice the velocity means twice as much momentum transferred per collision and twice as many collisions per unit time. * is the kinetic energy per particle ** = + + Therefore: : V_0 = \frac{V}{n} Therefore: : T = p \frac{V}{n} = p V_0 = m v^2 = 2 E Therefore temperature is twice the energy per degree of freedom per particle * T = 2 E Electromagnetism From Wikipedia:Lorentz–Heaviside units: Gravitoelectromagnetism :See also: Einstein_field_equations From Wikipedia:Gravitoelectromagnetism: According to general relativity, the gravitational field produced by a rotating object (or any rotating mass–energy) can, in a particular limiting case, be described by equations that have the same form as in classical electromagnetism. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak gravitational field or reasonably flat spacetime, the gravitational analogs to Maxwell's equations for electromagnetism, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in SI units are: where: * E'g is the static gravitational field (conventional gravity, also called ''gravitoelectric in analogous usage) in m⋅s−2; * '''E is the electric field; * B'''g is the gravitomagnetic field in s−1; * '''B is the magnetic field; * ρ''g is mass density in kg⋅m−3; * ''ρ is charge density: * J'g is mass current density or mass flux ('J'g = ''ρ''g'vρ, where v'''ρ is the velocity of the mass flow generating the gravitomagnetic field) in kg⋅m−2⋅s−1; * '''J is electric current density; * G'' is the gravitational constant in m3⋅kg−1⋅s−2; * ''ε''0 is the vacuum permittivity; * ''c is the speed of propagation of gravity (which is equal to the speed of light according to general relativity) in m⋅s−1. CGS From Wikipedia:Centimetre–gram–second system of units: References External links *The Spectrum of Riemannium *wisc.edu *Stoney Scale and Large Number Coincidences